Implementation of a Controller to Eliminate the Limit Cycle in the Inverted Pendulum on a Cart

<div>Polar plot of <svg height="12.7178pt" id="M80" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 31.8994 12.7178" width="31.8994pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"></path></g><g transform="matrix(.013,0,0,-0.013,8.892,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.39,0)"><path d="M394 607C394 634 378 655 350 655C315 655 294 620 294 594C294 564 314 546 339 546C372 546 394 574 394 607ZM360 349C373 414 374 448 351 448C321 448 249 404 183 338L197 313C223 336 265 367 276 367C285 367 283 355 275 313C241 129 220 23 189 -102C175 -158 153 -197 124 -197C106 -197 85 -187 70 -177C64 -173 51 -177 44 -183C34 -193 23 -209 23 -228C23 -244 46 -261 67 -261C86 -261 127 -245 181 -197C237 -147 279 -52 303 66L360 349Z"></path></g><g transform="matrix(.013,0,0,-0.013,18.89,0)"><path d="M615 290C615 392 563 448 529 448C511 448 491 441 484 434L483 425C518 400 546 349 546 273C546 158 504 46 417 46C381 46 356 77 356 142C356 184 380 293 392 334L387 340L307 323C305 288 296 228 287 188C265 94 221 52 180 52C144 52 112 89 112 171C112 278 164 372 241 430L226 454C106 394 23 288 23 155C23 40 79 -12 139 -12C184 -12 245 32 285 85C292 31 322 -12 376 -12C423 -12 470 18 501 45C570 104 615 184 615 290Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.186,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> and <span class="nowrap"><svg height="11.5564pt" id="M81" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 48.5496 11.5564" width="48.5496pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,7.631,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.871,0)"><path d="M368 703H309L44 -163H104L368 703Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.232,0)"><path d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"></path></g><g transform="matrix(.013,0,0,-0.013,30.142,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.64,0)"><path d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"></path></g><g transform="matrix(.013,0,0,-0.013,43.775,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg>.</span></div>

Complexity

fig6

Figure 6

Figure 6: Implementation of a Controller to Eliminate the Limit Cycle in the Inverted Pendulum on a Cart 